Going beyond variation of sets. (April 2017)

• Record Type: Journal Article
• Authors: Chlebík, Miroslav
• Is Part Of: Nonlinear analysis. Volume 153(2017)
• Page Start: 230
• Page End: 242
• Publication Date: 2017-04
• Subjects: primary 28A75 49Q15 -- secondary 26B15 28A78; Perimeter of sets -- Measure-theoretic boundary -- Integralgeometric measure; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.11.002
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study integralgeometric representations of variations of general sets A ⊂ R n without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function χ A is a signed Borel measure on R n with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss–Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a 'measure-theoretic boundary' plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of 'measure-theoretic boundary' and one can address the question to find notions of measure-theoretic boundary that are as fine as possible.